We investigate the monotone representation and measurability of generalized $\psi$-estimators introduced by the authors in 2022. Our first main result, applying the unique existence of a generalized $\psi$-estimator, allows us to construct this estimator in terms of a function $\psi$, which is decreasing in its second variable. We then interpret this result as a bridge from a nonconvex optimization problem to a convex one. Further, supposing that the underlying measurable space (sample space) has a measurable diagonal and some additional assumptions on $\psi$, we show that the measurability of a generalized $\psi$-estimator is equivalent to the measurability of the corresponding function $\psi$ in its first variable.

翻译：本文研究了作者于2022年提出的广义 $ψ$-估计量的单调表示与可测性问题。我们的第一个主要结果应用了广义 $ψ$-估计量的唯一存在性，使我们能够通过一个在其第二变量上递减的函数 $\psi$ 来构造该估计量。随后，我们将这一结果解释为从非凸优化问题到凸优化问题的一座桥梁。进一步地，假设基础可测空间（样本空间）具有可测对角线，并对 $\psi$ 施加若干附加条件，我们证明广义 $ψ$-估计量的可测性等价于相应函数 $\psi$ 在其第一变量上的可测性。